What is the maximum value o fx y subject to the condition x+y=8 ?

Options

A. 8
B. 16
C. 24
D. 32

Explanation

To find the maximum value of \( P = xy \) subject to the condition \( x + y = 8 \), we can follow these steps:

### Step 1: Express \( y \) in terms of \( x \)
Given the constraint \( x + y = 8 \), we can express \( y \) as:
\[ y = 8 - x \]

### Step 2: Substitute \( y \) in the product \( P \)
Now, substitute \( y \) in the product \( P \):
\[ P = x(8 - x) = 8x - x^2 \]

### Step 3: Differentiate \( P \) with respect to \( x \)
Next, we differentiate \( P \) to find the critical points:
\[ \frac{dP}{dx} = 8 - 2x \]

### Step 4: Set the derivative equal to zero
To find the maximum, set the derivative equal to zero:
\[ 8 - 2x = 0 \]
Solving for \( x \):
\[ 2x = 8 \quad \Rightarrow \quad x = 4 \]

### Step 5: Find \( y \) using the value of \( x \)
Now, substitute \( x = 4 \) back into the equation for \( y \):
\[ y = 8 - x = 8 - 4 = 4 \]

### Step 6: Calculate the maximum value of \( P \)
Now, substitute \( x \) and \( y \) back into the product \( P \):
\[ P = xy = 4 \times 4 = 16 \]

### Conclusion
Thus, the maximum value of \( P \) is:
\[ \boxed{16} \]

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